Problem 25 - Applied Calculus

Question

A company produces and sells shirts. The fixed costs are $$\$ 7000$$ and the variable costs are $$\$ 5$$ per shirt. (a) Shirts are sold for $$\$ 12$$ each. Find cost and revenue as functions of the quantity of shirts, \(q\). (b) The company is considering changing the selling price of the shirts. Demand is \(q=2000-40 p\), where \(p\) is price in dollars and \(q\) is the number of shirts. What quantity is sold at the current price of $$\$ 12 ?$$ What profit is realized at this price? (c) Use the demand equation to write cost and revenue as functions of the price, \(p\). Then write profit as a function of price. (d) Graph profit against price. Find the price that maximizes profits. What is this profit?

Step-by-Step Solution

Answer

The price of \(\$ 12\) maximizes profits, yielding a profit of \(\$ 3640\).

Step 1: Define Cost Function

The cost of producing shirts comprises fixed and variable costs. Fixed costs amount to \(\\( 7000\), and variable costs are \(\\) 5\) per shirt. Thus, the cost function \(C(q)\) is given by: \[ C(q) = 7000 + 5q \] where \(q\) is the number of shirts.

Step 2: Define Revenue Function

Revenue is earned by selling shirts at a price of \(\$ 12\) each. Hence, the revenue function \(R(q)\) is: \[ R(q) = 12q \] where \(q\) is the number of shirts sold.

Step 3: Determine Quantity Sold and Profit at Current Price

Given the demand equation \(q = 2000 - 40p\), substitute \(p = 12\):\[ q = 2000 - 40(12) = 2000 - 480 = 1520 \]To find profit, subtract total cost from revenue:Profit \(P(q)\) is given by \(P(q) = R(q) - C(q)\):\[ P(q) = 12q - (7000 + 5q) = 7q - 7000 \]At \(q = 1520\), \[ P(1520) = 7 \times 1520 - 7000 = 10640 - 7000 = 3640 \]So, the profit realized is \(\$ 3640\).

Step 4: Express Cost as Function of Price

Substitute the demand equation \(q = 2000 - 40p\) into the cost function \(C(q) = 7000 + 5q\):\[ C(p) = 7000 + 5(2000 - 40p) \]Simplify:\[ C(p) = 7000 + 10000 - 200p = 17000 - 200p \]

Step 5: Express Revenue as Function of Price

Substitute the demand equation \(q = 2000 - 40p\) into the revenue function \(R(q) = 12q\):\[ R(p) = 12(2000 - 40p) \]Simplify:\[ R(p) = 24000 - 480p \]

Step 6: Express Profit as Function of Price

Substitute the functions for cost and revenue in terms of price into the profit formula:\[ P(p) = R(p) - C(p) \]\[ P(p) = (24000 - 480p) - (17000 - 200p) \]Simplify:\[ P(p) = 24000 - 480p - 17000 + 200p \]\[ P(p) = 7000 - 280p \]

Step 7: Graph and Maximize Profit

Profit as a function of price is \(P(p) = 7000 - 280p\). This is a linear function with a negative slope, meaning that profit decreases as the price increases. Thus, the maximum profit occurs at the lowest price that still attracts demand.Set \(q = 0\) in \(2000 - 40p = 0\) to find when demand hits zero.\[ 40p = 2000 \]\[ p = 50 \]Hence, \(p > 0\) for prices less than \(\\( 50\) will maximize the profit. Since profit decreases with price, maximum profit at the lowest price with demand, which is \(p = 12\), yielding the profit computed earlier at \(\\) 3640 \).

Key Concepts

Cost Function

When a company produces goods, it incurs various costs associated with making those products. The cost function is a mathematical formula that helps represent these costs. In this exercise, the company has both fixed and variable costs.

• Fixed Costs are costs that do not change with the level of production. Here, the fixed costs amount to \(7000\) dollars, which includes expenses like rent, salaries, or equipment that are constant regardless of how many shirts are produced.
• Variable Costs vary with the production level. For each shirt made, there is an additional cost of \(5\) dollars. This includes costs of materials and labor that increase as more shirts are produced.

Combining these, the cost function for producing \(q\) shirts is given by the formula \(C(q) = 7000 + 5q\). This equation shows how total costs increase as more shirts are manufactured. The more shirts are produced, the greater the variable cost component, while the fixed cost remains constant.

Revenue Function

Revenue represents the income a company generates from selling its products or services. It depends on the price at which the products are sold and the quantity sold. In this situation, revenue comes from selling shirts.

• Each shirt is sold at \(12\) dollars.
• The revenue function is a simple multiplication of the price of the shirt by the number of shirts sold.

Thus, the revenue function, denoted as \(R(q)\), can be expressed as \(R(q) = 12q\), where \(q\) is the number of shirts sold. The revenue function is linear, meaning revenue grows consistently with additional sales. Understanding the revenue function helps businesses evaluate how much they stand to earn based on different sales volumes.

Demand Equation

In economics, the demand equation fundamentally illustrates how the quantity demanded of a good relates to its price. It helps predict consumer behavior regarding purchasing decisions at varying price points.
In this context, the demand equation is given by \(q = 2000 - 40p\), where \(q\) is the quantity sold and \(p\) represents the price. This equation has several implications:

• The negative coefficient of \(p\) (-40) indicates an inverse relationship between price and quantity demanded—when the price increases, the quantity demanded tends to decrease.
• The leading term of 2000 suggests that if the price were \(0\), theoretically, the quantity demanded would be \(2000\) shirts.

By substituting different prices into this equation, businesses can predict how many units they're likely to sell, helping them make informed pricing decisions for optimizing sales and maximizing profits. In this exercise, substituting the current price of \(12\) dollars into the demand equation results in \(q = 1520\), meaning 1520 shirts can be sold at this price point, which informs the projected revenue and profits at this price level.